cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A354135 Expansion of e.g.f. 1/(1 - log(1 + x)^5/120).

Original entry on oeis.org

1, 0, 0, 0, 0, 1, -15, 175, -1960, 22449, -269073, 3403070, -45510630, 643152796, -9586136560, 150319669136, -2473024029840, 42562037379744, -764017130370276, 14260496108114340, -275877454002406236, 5512350021871343616, -113318466860425703184
Offset: 0

Views

Author

Seiichi Manyama, May 18 2022

Keywords

Crossrefs

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-log(1+x)^5/120)))
    
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, binomial(i, j)*stirling(j, 5, 1)*v[i-j+1])); v;
    
  • PARI
    a(n) = sum(k=0, n\5, (5*k)!*stirling(n, 5*k, 1)/120^k);

Formula

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * Stirling1(k,5) * a(n-k).
a(n) = Sum_{k=0..floor(n/5)} (5*k)! * Stirling1(n,5*k)/120^k.

A354229 Expansion of e.g.f. 1/(1 - log(1 + x)^3).

Original entry on oeis.org

1, 0, 0, 6, -36, 210, -630, -5376, 153048, -2194296, 22190760, -93956544, -2677330656, 97821857952, -2019503487456, 27899293618944, -98409183995520, -9548919666829440, 410311098024923520, -10652005874894469120, 176525303194482117120, -46197517147757867520
Offset: 0

Views

Author

Seiichi Manyama, May 20 2022

Keywords

Crossrefs

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-log(1+x)^3)))
    
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=6*sum(j=1, i, binomial(i, j)*stirling(j, 3, 1)*v[i-j+1])); v;
    
  • PARI
    a(n) = sum(k=0, n\3, (3*k)!*stirling(n, 3*k, 1));

Formula

a(0) = 1; a(n) = 6 * Sum_{k=1..n} binomial(n,k) * Stirling1(k,3) * a(n-k).
a(n) = Sum_{k=0..floor(n/3)} (3*k)! * Stirling1(n,3*k).

A354136 Expansion of e.g.f. exp(log(1 + x)^3/6).

Original entry on oeis.org

1, 0, 0, 1, -6, 35, -215, 1414, -9912, 73044, -552570, 4102626, -26654826, 79506492, 2154425364, -73527421176, 1708053626880, -35961691589640, 736338276883080, -15067241745943680, 312009998091705720, -6579362641255341120, 141704946709227843480
Offset: 0

Views

Author

Seiichi Manyama, May 18 2022

Keywords

Crossrefs

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(log(1+x)^3/6)))
    
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, binomial(i-1, j-1)*stirling(j, 3, 1)*v[i-j+1])); v;
    
  • PARI
    a(n) = sum(k=0, n\3, (3*k)!*stirling(n, 3*k, 1)/(6^k*k!));

Formula

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * Stirling1(k,3) * a(n-k).
a(n) = Sum_{k=0..floor(n/3)} (3*k)! * Stirling1(n,3*k)/(6^k * k!).
Showing 1-3 of 3 results.